Foundations of a set theory for fuzzy sets. 40 years of development

Besides their importance for applications, fuzzy sets as generalized sets call for a set theoretic understanding. This paper concisely surveys the different types of approaches toward this foundational problem which have been offered during the 40 years since fuzzy sets first have appeared in print.

[1]  Nando Prati.  An axiomatization of fuzzy classes. , 1988 .

[2]  Thoralf Skolem,et al.  Bemerkungen zum Komprehensionsaxiom. Dem Andenken an Heinrich Scholz gewidmet , 1957 .

[3]  Petr Cintula,et al.  Product Ł ukasiewicz Logic , 2004, Arch. Math. Log..

[4]  Lotfi A. Zadeh,et al.  Fuzzy Sets , 1996, Inf. Control..

[5]  U. Höhle.  Presheaves over GL-monoids , 1995 .

[6]  U. Höhle.  M-valued Sets and Sheaves over Integral Commutative CL-Monoids , 1992 .

[7]  Joseph A. Goguen,et al.  Concept Representation in Natural and Artificial Languages: Axioms, Extensions and Applications for Fuzzy Sets , 1974, Int. J. Man Mach. Stud..

[8]  Petr Hájek.  On arithmetic in the Cantor- Łukasiewicz fuzzy set theory , 2005, Arch. Math. Log..

[9]  D. Çoker,et al.  On the axiomatic theory of fuzzy sets , 1993 .

[10]  August J Weidner.  Fuzzy sets and Boolean-valued universes , 1981 .

[11]  J. Goguen.  L-fuzzy sets , 1967 .

[12]  Oswald Wyler,et al.  Fuzzy logic and categories of fuzzy sets , 1995 .

[13]  F W Lawvere,et al.  AN ELEMENTARY THEORY OF THE CATEGORY OF SETS. , 1964, Proceedings of the National Academy of Sciences of the United States of America.

[14]  D. Scott,et al.  Sheaves and logic , 1979 .

[15]  Renatus Ziegler,et al.  On the Foundations of Set Theory , 1996 .

[16]  John L. Bell,et al.  Boolean-valued models and independence proofs in set theory , 1977 .

[17]  Review: F. William Lawvere, An Elementary Theory of the Category of sets , 1972 .

[18]  N. Prati.  About the axiomatizations of fuzzy sets theory , 1991 .

[19]  S. Gottwald.  Set theory for fuzzy sets of higher level , 1979 .

[20]  Archive for mathematical logic , 1987 .

[21]  Joseph G. Brown,et al.  A Note on Fuzzy Sets , 1971, Inf. Control..

[22]  W. D. Blizard.  Real-values multisets and fuzzy sets , 1989 .

[23]  Siegfried Gottwald,et al.  Fuzzy Sets and Fuzzy Logic , 1993 .

[24]  Siegfried Gottwald,et al.  Fuzzy uniqueness of fuzzy mappings , 1980 .

[25]  Zhang Jinwen,et al.  A unified treatment of fuzzy set theory and Boolean-valued set theory—Fuzzy set structures and normal fuzzy set structures , 1980 .

[26]  Petr Cintula,et al.  From fuzzy logic to fuzzy mathematics: A methodological manifesto , 2006, Fuzzy Sets Syst..

[27]  E. William Chapin.  Set-valued set theory. II , 1975, Notre Dame J. Formal Log..

[28]  U. Höhle.  Commutative, residuated 1—monoids , 1995 .

[29]  Petr Hájek,et al.  A set theory within fuzzy logic , 2001, Proceedings 31st IEEE International Symposium on Multiple-Valued Logic.

[30]  M. Eytan,et al.  Fuzzy sets: A topos-logical point of view , 1981 .

[31]  Nando Prati,et al.  On the comparison between fuzzy set axiomatizations , 1992 .

[32]  John Lake,et al.  Sets, Fuzzy Sets, Multisets and Functions , 1976 .

[33]  Lotfi A. Zadeh,et al.  Please Scroll down for Article International Journal of General Systems Fuzzy Sets and Systems* Fuzzy Sets and Systems* , 2022 .

[34]  S. Gottwald.  A Treatise on Many-Valued Logics , 2001 .

[35]  Satoko Titani,et al.  Fuzzy logic and fuzzy set theory , 1992, Arch. Math. Log..

[36]  Satoko Titani,et al.  Intuitionistic fuzzy logic and intuitionistic fuzzy set theory , 1984, Journal of Symbolic Logic.

[37]  Vilém Novák.  An attempt at gödel-bernays-like axiomatization of fuzzy sets , 1980 .

[38]  Siegfried Gottwald,et al.  Universes of fuzzy sets-a short survey , 2003, 33rd International Symposium on Multiple-Valued Logic, 2003. Proceedings..

[39]  D. Scott.  Identity and existence in intuitionistic logic , 1979 .

[40]  Abraham Kandel,et al.  An axiomatic approach to fuzzy set theory , 1990, Inf. Sci..

[41]  Petr Cintula,et al.  Fuzzy class theory , 2005, Fuzzy Sets Syst..

[42]  E. William Chapin.  Set-valued set theory. I , 1974, Notre Dame J. Formal Log..